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Theorem ovresd 7599
Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
ovresd.1 (𝜑𝐴𝑋)
ovresd.2 (𝜑𝐵𝑋)
Assertion
Ref Expression
ovresd (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))

Proof of Theorem ovresd
StepHypRef Expression
1 ovresd.1 . 2 (𝜑𝐴𝑋)
2 ovresd.2 . 2 (𝜑𝐵𝑋)
3 ovres 7598 . 2 ((𝐴𝑋𝐵𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
41, 2, 3syl2anc 584 1 (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105   × cxp 5686  cres 5690  (class class class)co 7430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-res 5700  df-iota 6515  df-fv 6570  df-ov 7433
This theorem is referenced by:  sscres  17870  fullsubc  17900  fullresc  17901  funcres2c  17954  rngchom  20639  ringchom  20668  rhmsubclem4  20704  irinitoringc  21507  psmetres2  24339  xmetres2  24386  prdsdsf  24392  xpsdsval  24406  xmssym  24490  xmstri2  24491  mstri2  24492  xmstri  24493  mstri  24494  xmstri3  24495  mstri3  24496  msrtri  24497  tmsxpsval  24566  ngptgp  24664  nlmvscn  24723  nrginvrcn  24728  nghmcn  24781  cnmpt1ds  24877  cnmpt2ds  24878  ipcn  25293  caussi  25344  causs  25345  minveclem2  25473  minveclem3b  25475  minveclem3  25476  minveclem4  25479  minveclem6  25481  ftc1lem6  26096  ulmdvlem1  26457  abelth  26499  cxpcn3  26805  rlimcnp  27022  hhssnv  31292  madjusmdetlem3  33789  qqhcn  33953  qqhucn  33954  ftc1cnnc  37678  ismtyres  37794  isdrngo2  37944  naddcnffo  43353
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